Question

Which of the following complex functions is/are analytic on the complex plane?

• A. \( f(z) = j Re(z) \)
  (B) \( f(z) = Im(z) \)
  (C) \( f(z) = e^{|z|} \)
  (D) \( f(z) = z^2 - z \)
• B.
• C.
• D.

Hint:

A complex function is analytic if it satisfies the Cauchy-Riemann equations and its derivatives are continuous in the given domain.

Answer 1

The Correct Option is D

From the given question, \[ f(z) = z^2 - z \tag{i} \] Substituting the value of \(z\) into the above equation: \[ z = (x + iy) \] we get: \[ z = (x^2 - y^2 + i(2xy)) - x - iy \] \[ = (x^2 - y^2 - x) + i(2xy - y) \] Thus, \[ u = x^2 - y^2 - x, \quad v = 2xy - y \] Now, applying the Cauchy-Riemann (C-R) equations: \[ u_x = v_y \] \[ u_y = -v_x \] we have: \[ u_x = 2x - 1, \quad v_y = 2y \] \[ u_y = -2y, \quad v_x = 2x - 1 \] Substituting: \[ (2x - 1) = (2x - 1) \quad \text{and} \quad -2y = -(2y) \] This shows that the C-R equations are satisfied. Hence, the function \(f(z) = z^2 - z\) is analytic on the complex plane. Hence, the correct option is (D).